Rabin signature algorithm

The Rabin signature scheme is parametrized by a randomized hash function ${\displaystyle H(m,u)}$ of a message ${\displaystyle m}$ and ${\displaystyle k}$-bit randomization string ${\displaystyle u}$.

Public key

A public key is a pair of integers ${\displaystyle (n,b)}$ with ${\displaystyle 0\leq b<n}$ and ${\displaystyle n}$ odd.

Signature

A signature on a message ${\displaystyle m}$ is a pair ${\displaystyle (u,x)}$ of a ${\displaystyle k}$-bit string ${\displaystyle u}$ and an integer ${\displaystyle x}$ such that

$${\displaystyle x(x+b)\equiv H(m,u){\pmod {n}}.}$$

Private key

The private key for a public key ${\displaystyle (n,b)}$ is the secret odd prime factorization ${\displaystyle p\cdot q}$ of ${\displaystyle n}$, chosen uniformly at random from some space of large primes. Let ${\displaystyle d=(b/2){\bmod {n}}}$, ${\displaystyle d_{p}=(b/2){\bmod {p}}}$, and ${\displaystyle d_{q}=(b/2){\bmod {q}}}$. To make a signature on a message ${\displaystyle m}$, the signer picks a ${\displaystyle k}$-bit string ${\displaystyle u}$ uniformly at random, and computes ${\displaystyle c:=H(m,u)}$. If ${\displaystyle c+d^{2}}$ is a quadratic nonresidue modulo ${\displaystyle n}$, then the signer throws away ${\displaystyle u}$ and tries again. Otherwise, the signer computes

$${\displaystyle {\begin{aligned}x_{p}&:={\Bigl (}-d_{p}\pm {\sqrt {c+{d_{p}}^{2}}}{\Bigr )}{\bmod {p}}\\x_{q}&:={\Bigl (}-d_{q}\pm {\sqrt {c+{d_{q}}^{2}}}{\Bigr )}{\bmod {q}},\end{aligned}}}$$

using a standard algorithm for computing square roots modulo a prime—picking ${\displaystyle p\equiv q\equiv 3{\pmod {4}}}$ makes it easiest. Square roots are not unique, and different variants of the signature scheme make different choices of square root;[4] in any case, the signer must ensure not to reveal two different roots for the same hash ${\displaystyle c}$. The signer then uses the Chinese remainder theorem to solve the system

$${\displaystyle {\begin{aligned}x&\equiv x_{q}{\pmod {n}}\\x&\equiv x_{p}{\pmod {n}}\end{aligned}}}$$

for ${\displaystyle x}$. The signer finally reveals ${\displaystyle (u,x)}$.

Correctness of the signing procedure follows by evaluating ${\displaystyle x(x+b)-H(m,u)}$ modulo ${\displaystyle p}$ and ${\displaystyle q}$ with ${\displaystyle x}$ as constructed. For example, in the simple case where ${\displaystyle b=0}$, ${\displaystyle x}$ is simply a square root of ${\displaystyle H(m,u)}$ modulo ${\displaystyle n}$. The number of trials for ${\displaystyle u}$ is geometrically distributed with expectation around 4, because about 1/4 of all integers are quadratic residues modulo ${\displaystyle n}$.

Security against any adversary defined generically in terms of a hash function ${\displaystyle H}$ (i.e., security in the random oracle model) follows from the difficulty of factoring ${\displaystyle n}$: Any such adversary with high probability of success at forgery can, with nearly as high probability, find two distinct square roots ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ of a random integer ${\displaystyle c}$ modulo ${\displaystyle n}$. If ${\displaystyle x_{1}\pm x_{2}\not \equiv 0{\pmod {n}}}$ then ${\displaystyle \gcd(x_{1}\pm x_{2},n)}$ is a nontrivial factor of ${\displaystyle n}$, since ${\displaystyle {x_{1}}^{2}\equiv {x_{2}}^{2}\equiv c{\pmod {n}}}$ so ${\displaystyle n\mid {x_{1}}^{2}-{x_{2}}^{2}=(x_{1}+x_{2})(x_{1}-x_{2})}$ but ${\displaystyle n\nmid x_{1}\pm x_{2}}$.[3] Formalizing the security in modern terms requires filling in some additional details, such as the codomain of ${\displaystyle H}$; if we set a standard size ${\displaystyle K}$ for the prime factors, ${\displaystyle 2^{K-1}<p<q<2^{K}}$, then we might specify ${\displaystyle H\colon \{0,1\}^{*}\times \{0,1\}^{k}\to \{0,1\}^{K}}$.[5]

Randomization of the hash function was introduced to allow the signer to find a quadratic residue, but randomized hashing for signatures later became relevant in its own right for tighter security theorems[3] and resilience to collision attacks on fixed hash functions.[7][8][9]

The quantity ${\displaystyle b}$ in the public key adds no security, since any algorithm to solve congruences ${\displaystyle x(x+b)\equiv c{\pmod {n}}}$ for ${\displaystyle x}$ given ${\displaystyle b}$ and ${\displaystyle c}$ can be trivially used as a subroutine in an algorithm to compute square roots modulo ${\displaystyle n}$ and vice versa, so implementations can safely set ${\displaystyle b=0}$ for simplicity; ${\displaystyle b}$ was discarded altogether in treatments after the initial proposal.[10][3][6][4]

The Rabin signature scheme was later tweaked by Williams in 1980[10] to choose ${\displaystyle p\equiv 3{\pmod {8}}}$ and ${\displaystyle q\equiv 7{\pmod {8}}}$, and replace a square root ${\displaystyle x}$ by a tweaked square root ${\displaystyle (e,f,x)}$, with ${\displaystyle e=\pm 1}$ and ${\displaystyle f\in \{1,2\}}$, so that a signature instead satisfies

$${\displaystyle efx^{2}\equiv H(m,u){\pmod {n}},}$$

which allows the signer to create a signature in a single trial without sacrificing security. This variant is known as Rabin–Williams.[4][6] Further variants allow tradeoffs between signature size and verification speed, partial message recovery, signature compression, and public key compression.[4]

Variants without the hash function have been published in textbooks,[11][12] crediting Rabin for exponent 2 but not for the use of a hash function. These variants are trivially broken—for example, the signature ${\displaystyle x=2}$ can be forged by anyone as a valid signature on the message ${\displaystyle m=4}$ if the signature verification equation is ${\displaystyle x^{2}\equiv m{\pmod {n}}}$ instead of ${\displaystyle x^{2}\equiv H(m,u){\pmod {n}}}$.

In the original paper,[2] the hash function ${\displaystyle H(m,u)}$ was written with the notation ${\displaystyle C(MU)}$, with C for compression, and using juxtaposition to denote concatenation of ${\displaystyle M}$ and ${\displaystyle U}$ as bit strings:

By convention, when wishing to sign a given message, ${\displaystyle M}$, [the signer] ${\displaystyle P}$ adds as suffix a word ${\displaystyle U}$ of an agreed upon length ${\displaystyle k}$. The choice of ${\displaystyle U}$ is randomized each time a message is to be signed. The signer now compresses ${\displaystyle M_{1}=MU}$ by a hashing function to a word ${\displaystyle C(M_{1})=c}$, so that as a binary number ${\displaystyle c\leq n}$…

This notation has led to some confusion among some authors later who ignored the ${\displaystyle C}$ part and misunderstood ${\displaystyle MU}$ to mean multiplication, giving the misapprehension of a trivially broken signature scheme.[13]

• Rabin–Williams signatures at cr.yp.to